On the general form of linear functionals on the Hardy spaces $H^1$ over compact Abelian groups and some of its applications

TL;DR

This paper generalizes Fefferman's theorems on linear functionals in Hardy spaces from the circle group to arbitrary compact Abelian groups with totally ordered duals, with applications to Fourier series and atomic decompositions.

Contribution

It extends the characterization of linear functionals on $H^1$ to a broader class of groups, providing new tools for harmonic analysis on these groups.

Findings

Generalization of Fefferman's theorems to compact Abelian groups

Applications to lacunary Fourier series

Results on atomic decomposition on the 2D torus

Abstract

The celebrated Fefferman's theorems on the general form of linear functionals on the Hardy space $H^1$ over the circle group is generalized to the case of an arbitrary compact Abelian group with totally ordered dual. Several corollaries that can be applied to multiple lacunary Fourier series and atomic decomposition on the two dimensional torus are obtained.